Home A class of quaternionic Fourier orthonormal bases
Article
Licensed
Unlicensed Requires Authentication

A class of quaternionic Fourier orthonormal bases

  • Yun-Zhang Li EMAIL logo and Xiao-Li Zhang
Published/Copyright: October 4, 2023

Abstract

Due to its applications in signal analysis and image processing, the quaternionic Fourier analysis has received increasing attention. In particular, quaternionic Gabor frames (QGFs) attracted some mathematicians’ interest. From the literatures, some results on QGFs are based on quaternionic Fourier orthonormal bases. But those used so-called quaternionic Fourier orthonormal bases have a gap that they are all incomplete. In this paper, we present a class of quaternionic Fourier orthonormal bases, and using them derive the corresponding Gabor orthonormal bases.


Communicated by Christopher D. Sogge


Award Identifier / Grant number: 12371091

Award Identifier / Grant number: 11971043

Funding statement: This work was supported by National Natural Science Foundation of China (Grants No. 12371091, No. 11971043).

References

[1] S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields, Internat. Ser. Monogr. Phys. 88, Oxford University, New York, 1995. 10.1093/oso/9780195066432.003.0002Search in Google Scholar

[2] L. Akila and R. Roopkumar, Multidimensional quaternionic Gabor transforms, Adv. Appl. Clifford Algebr. 26 (2016), no. 3, 985–1011. 10.1007/s00006-015-0634-xSearch in Google Scholar

[3] M. Bahri, E. S. M. Hitzer, R. Ashino and R. Vaillancourt, Windowed Fourier transform of two-dimensional quaternionic signals, Appl. Math. Comput. 216 (2010), no. 8, 2366–2379. 10.1016/j.amc.2010.03.082Search in Google Scholar

[4] M. Bahri, E. S. M. Hitzer, A. Hayashi and R. Ashino, An uncertainty principle for quaternion Fourier transform, Comput. Math. Appl. 56 (2008), no. 9, 2398–2410. 10.1016/j.camwa.2008.05.032Search in Google Scholar

[5] P. Bas, N. Le Bihan and J. M. Chassery, Color image watermarking using quaternion Fourier transform, Proceedings of the IEEE International Conference on Acoustics Speech and Signal Processing, ICASSP, Hong-Kong (2003), 521–524. 10.1109/ICASSP.2003.1199526Search in Google Scholar

[6] E. Bayro-Corrochano, N. Trujillo and M. Naranjo, Quaternion Fourier descriptors for the preprocessing and recognition of spoken words using images of spatiotemporal representations, J. Math. Imaging Vision 28 (2007), no. 2, 179–190. 10.1007/s10851-007-0004-ySearch in Google Scholar

[7] G. Birkhoff and J. von Neumann, The logic of quantum mechanics, Ann. of Math. (2) 37 (1936), no. 4, 823–843. 10.2307/1968621Search in Google Scholar

[8] T. Bülow, Hypercomplex spectral signal representations for the processing and analysis of images, Ph.D. Thesis, Institut für Informatik und Praktische Mathematik, University of Kiel, Kiel, 1999. Search in Google Scholar

[9] P. Cerejeiras, S. Hartmann and H. Orelma, Structural results for quaternionic Gabor frames, Adv. Appl. Clifford Algebr. 28 (2018), no. 5, Paper No. 86. 10.1007/s00006-018-0901-8Search in Google Scholar

[10] M. El Kassimi and S. Fahlaoui, The two-sided Gabor quaternionic Fourier transform and uncertainty principles, Recent Advances in Mathematics and Technology, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham (2020), 3–19. 10.1007/978-3-030-35202-8_1Search in Google Scholar

[11] T. A. Ell, Hypercomplex spectral transformations, Ph.D. Dissertation, University of Minnesota, 1992. Search in Google Scholar

[12] T. A. Ell, Quaternion-fourier transforms for analysis of two-dimensional linear time-invariant partial differential systems, Proceeding of the 32nd Conference on Decision and Control (San Antonio 1993), IEE Press, Piscataway (1993), 1830–1841. 10.1109/CDC.1993.325510Search in Google Scholar

[13] T. A. Ell and S. J. Sangwine, Hypercomplex Fourier transforms of color images, IEEE Trans. Image Process. 16 (2007), no. 1, 22–35. 10.1109/TIP.2006.884955Search in Google Scholar PubMed

[14] M. Felsberg, Low-Level image processing with the structure multivector, Ph.D. Thesis, Institut für Informatik und Praktische Mathematik, University of Kiel, Kiel, 2002. Search in Google Scholar

[15] Y. Fu, U. Kähler and P. Cerejeiras, the Balian–Low theorem for the windowed quaternionic Fourier transform, Adv. Appl. Clifford Algebr. 22 (2012), no. 4, 1025–1040. 10.1007/s00006-012-0324-xSearch in Google Scholar

[16] K. Gürlebeck, K. Habetha and W. Sprößig, Funktionentheorie in der Ebene und im Raum, Birkhäuser, Basel, 2006. Search in Google Scholar

[17] S. Hartmann, Quaternionic Gabor Expansion, Technische Universität Bergakademie Freiberg, Freiberg, 2015. Search in Google Scholar

[18] S. Hartmann, Some results on the lattice parameters of quaternionic Gabor frames, Adv. Appl. Clifford Algebr. 26 (2016), no. 1, 137–149. 10.1007/s00006-015-0587-0Search in Google Scholar

[19] S. Hartmann, Relaxed quaternionic Gabor expansions at critical density, Math. Methods Appl. Sci. 40 (2017), no. 5, 1666–1678. 10.1002/mma.4087Search in Google Scholar

[20] E. Hitzer, The quaternion domain Fourier transform and its properties, Adv. Appl. Clifford Algebr. 26 (2016), no. 3, 969–984. 10.1007/s00006-015-0620-3Search in Google Scholar

[21] E. M. S. Hitzer, Quaternion Fourier transform on quaternion fields and generalizations, Adv. Appl. Clifford Algebr. 17 (2007), no. 3, 497–517. 10.1007/s00006-007-0037-8Search in Google Scholar

[22] B. Kamel and E. Tefjeni, Uncertainty principle for the two-sided quaternion windowed Fourier transform, Integral Transforms Spec. Funct. 30 (2019), no. 5, 362–382. 10.1080/10652469.2019.1572138Search in Google Scholar

[23] J. Li and J. He, Some results for the two-sided quaternionic Gabor Fourier transform and quaternionic Gabor frame operator, Adv. Appl. Clifford Algebr. 31 (2021), no. 1, Paper No. 1. 10.1007/s00006-020-01101-8Search in Google Scholar

[24] H. Mejjaoli, Paley–Wiener theorems for the two-sided quaternion Fourier transform, Adv. Appl. Clifford Algebr. 27 (2017), no. 2, 1611–1631. 10.1007/s00006-016-0699-1Search in Google Scholar

[25] S.-C. Pei, J.-J. Ding and J.-H. Chang, Efficient implementation of quaternion Fourier transform, convolution, and correlation by 2-D complex FFT, IEEE Trans. Signal Process. 49 (2001), no. 11, 2783–2797. 10.1109/78.960426Search in Google Scholar

[26] S. J. Sangwine, Fourier transforms of color images using quaternion or hypercomplex numbers, Electron. Lett. 32 (1996), no. 21, 1979–1980. 10.1049/el:19961331Search in Google Scholar

[27] M. Shapiro and L. M. Tovar, On a class of integral representations related to the two-dimensional Helmholtz operator, Operator Theory for Complex and Hypercomplex Analysis, Contemp. Math. 212, American Mathematical Society, Providence (1998), 229–244. 10.1090/conm/212/02886Search in Google Scholar

Received: 2023-05-21
Published Online: 2023-10-04
Published in Print: 2024-05-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 10.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2023-0190/html
Scroll to top button